Mathematics · 12th Grade · Transcendental Functions and Growth · Weeks 1-9

Applications of e and Natural Logarithms

Solving real-world problems involving continuous growth, decay, and related rates.

• Common Core State Standards
• CCSS.Math.Content.HSF.LE.A.4

About This Topic

The applications of e and natural logarithms span some of the most important quantitative reasoning students will encounter after high school: continuous population growth, radioactive decay, Newton's law of cooling, and pharmacokinetic drug concentration models. Each scenario is governed by the differential equation dy/dt = ky, whose solution is always y = y_0 * e^(kt). Students who recognize this structure can immediately write a model, interpret its parameters, and answer questions about the system without starting from scratch each time.

A key learning target here is parameter interpretation: k > 0 signals growth, k < 0 signals decay, and the magnitude of k controls how fast. Changing k and observing the graph -- rather than just noting its value in a formula -- builds the intuitive sense of scale that makes these models useful. CCSS.Math.Content.HSF.LE.A.4 places this work in the context of expressing exponential models and finding parameters from data.

Active learning is particularly valuable because authentic context is readily available. Students who build their own decay model from actual half-life data, or compare continuous and discrete drug dosing schedules for the same medication, are doing genuine scientific reasoning. These experiences build quantitative literacy that goes well beyond passing an exam.

Key Questions

1. Design a model using e and natural logarithms to represent a continuous process.
2. Justify the choice of an exponential or logarithmic model for a given real-world scenario.
3. Evaluate the impact of changing parameters in a continuous growth model.

Learning Objectives

• Design a mathematical model using the base of the natural logarithm, e, to represent continuous growth or decay scenarios.
• Analyze the impact of parameter changes (growth rate k, initial value y_0) on the behavior of exponential growth and decay functions.
• Evaluate the appropriateness of using continuous exponential models versus discrete models for specific real-world phenomena.
• Calculate the time required for a quantity to double or halve given a continuous growth or decay rate.
• Explain the relationship between a rate of change and its corresponding exponential growth or decay model.

Before You Start

Solving Exponential Equations

Why: Students need to be proficient in isolating variables within exponential expressions, often using logarithms, to solve for time or rates.

Properties of Logarithms

Why: A strong understanding of logarithm properties is essential for manipulating and simplifying exponential and logarithmic equations encountered in these models.

Basic Exponential Functions

Why: Familiarity with the general form and behavior of exponential functions, including growth and decay, provides a foundation for understanding the specific continuous models.

Key Vocabulary

Continuous Growth/Decay	A process where a quantity changes at a rate proportional to its current value, modeled by functions involving the base of the natural logarithm, e.
Exponential Growth Model	A function of the form y = y_0 * e^(kt), where y_0 is the initial amount, k is the continuous growth rate (k > 0), and t is time.
Exponential Decay Model	A function of the form y = y_0 * e^(kt), where y_0 is the initial amount, k is the continuous decay rate (k < 0), and t is time.
Half-life	The time it takes for a decaying substance to reduce to half of its initial amount, often modeled using exponential decay.

Watch Out for These Misconceptions

Common Misconception

Any growth or decay problem can be modeled with y = a * b^t; e-based models are just an alternative form.

What to Teach Instead

Both forms represent the same family of functions, but the e-based form is preferred when k is a continuous rate (like a continuously compounded interest rate or a decay constant). Students who understand continuous vs. discrete growth can choose the appropriate form and interpret its parameters correctly, rather than treating both as interchangeable black boxes.

Common Misconception

The half-life is the time for the quantity to reach zero.

What to Teach Instead

An exponential decay function approaches zero asymptotically -- it never actually reaches it. The half-life is the time for the quantity to halve, not to vanish. After n half-lives, (1/2)^n of the original remains. Graphing the decay curve and tracing successive half-lives visually makes this asymptotic behavior clear.

Active Learning Ideas

Modeling Workshop: Build Your Own Decay Model

Groups choose a radioactive isotope (carbon-14, iodine-131, or uranium-238) from a reference card, look up its half-life, and derive the decay constant k. They then write the full model, graph it, and answer three specific questions: when is 10% remaining, what fraction remains after 5 half-lives, and what was the initial amount if 2g remains after a given time.

35 min·Small Groups

Think-Pair-Share: Choosing the Right Model

Students are given five real-world scenarios (population growth, cooling coffee, bacterial culture, drug dosing, nuclear waste storage) and must decide with a partner whether continuous or discrete modeling is more appropriate and what sign k should have. Pairs share their reasoning and the class resolves any disagreements.

18 min·Pairs

Parameter Investigation: What Does k Really Control?

Using a graphing calculator or Desmos, small groups graph y = 1000 * e^(kt) for k = 0.1, 0.5, 1.0, -0.1, -0.5, and -1.0. For each, they record the doubling or halving time, then derive the general formula t_double = ln(2)/k. Groups present their derivation and confirm the formula works across all cases.

28 min·Small Groups

Real-World Connections

• Biologists use continuous growth models to predict population dynamics for species like bacteria or invasive plants, estimating how quickly populations will expand under ideal conditions.
• Pharmacists and medical researchers utilize exponential decay models to determine drug dosages and understand how quickly a medication is eliminated from the body, ensuring therapeutic levels are maintained.
• Geologists apply radioactive decay models, which are continuous exponential processes, to date ancient rocks and artifacts, providing insights into Earth's history and archaeological findings.

Assessment Ideas

Quick Check

Provide students with a scenario, such as 'A bacterial culture starts with 100 cells and grows continuously at a rate of 15% per hour.' Ask them to write the specific exponential growth model and calculate the number of cells after 3 hours.

Discussion Prompt

Present two scenarios: one for compound interest compounded annually and another for continuously compounded interest. Ask students: 'Which scenario will result in more money over 10 years, and why? How does the continuous model differ from the discrete model in representing financial growth?'

Exit Ticket

Give students a half-life value for a radioactive isotope (e.g., Carbon-14 has a half-life of 5730 years). Ask them to write the decay model and determine what percentage of the original isotope remains after 11,460 years.

Frequently Asked Questions

What is the general formula for continuous exponential growth and decay?

The general model is y = y_0 * e^(kt), where y_0 is the initial amount, k is the continuous growth (k > 0) or decay (k < 0) rate, and t is time. This formula solves the differential equation dy/dt = ky, which describes any quantity that changes at a rate proportional to its current amount.

How do you find the decay constant k from a half-life?

Set y = y_0/2 and t = half-life in the decay formula. Dividing both sides by y_0 gives 1/2 = e^(k * t_half). Taking the natural log of both sides and solving yields k = ln(1/2) / t_half = -ln(2) / t_half. The negative sign confirms k is a decay rate.

How is Newton's law of cooling modeled with e?

Newton's law of cooling states that an object's temperature approaches the ambient temperature at a rate proportional to the current temperature difference. The model is T(t) = T_ambient + (T_0 - T_ambient) * e^(kt), where k is negative. The object cools quickly at first and slows as it approaches room temperature.

How does active learning help students apply e-based models to real problems?

Building a model from scratch -- choosing a scenario, deriving k from known data, writing the equation, and answering specific questions -- develops a transferable problem-solving process. Students who have done this in a small group setting can replicate the process independently because they built the model themselves rather than copying a worked example.